Question: The graph of a sinusoidal function has a maximum point at $(0,10)$ and then intersects its midline at $\left(\dfrac{\pi}{4},4\right)$. Write the formula of the function, where $x$ is entered in radians. $f(x)=$
The strategy First, let's use the given information to determine the function's amplitude, midline, and period. Then, we should determine whether to use a sine or a cosine function, based on the point where $x=0$. Finally, we should determine the parameters of the function's formula by considering all the above. Determining the amplitude, midline, and period The midline intersection is at $y={4}$, so this is the midline. The maximum point is $6$ units above the midline, so the amplitude is ${6}$. The midline intersection is $\dfrac{\pi}{4}$ units to the right of the maximum point, so the period is $4\cdot \dfrac{\pi}{4}={\pi}$. [Why did we multiply by 4?] Determining the type of function to use Since the graph has an extremum point at $x=0$, we should use the cosine function and not the sine function. This means there's no horizontal shift, so the function is of the form $a\cos(bx)+d$. [How do we know that?] Determining the parameters in $a\cos(bx)+d$ Since the extremum point at $x=0$ is a maximum point, we know that $a>0$. [How do we know that?] The amplitude is ${6}$, so $|a|={6}$. Since $a>0$, we can conclude that $a=6$. The midline is $y={4}$, so $d=4$. The period is ${\pi}$, so $b=\dfrac{2\pi}{{\pi}}=2$. The answer $f(x)=6\cos\left(2x\right)+4$